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### Course: Precalculus>Unit 7

Lesson 8: Using matrices to transform the plane

# Using matrices to transform the plane: Mapping a vector

2X2 matrices can define transformations for the entire plane. In this worked example, we see how to find the image of a given vector under the transformation defined by a given matrix. Created by Sal Khan.

## Want to join the conversation?

• It looks like this has already been outlined in the previous section. In "Matrices as transformations of the plane".
(1 vote)
• Yes, you are correct. The notation and concept of using matrices to represent transformations of vectors was already discussed in the previous section "Matrices as transformations of the plane". In that section, we saw that we can use a matrix to represent a linear transformation of a vector in a two-dimensional space.

Specifically, we can use a transformation matrix to map each point in the original space to a corresponding point in the transformed space. This transformation is performed by multiplying the transformation matrix by the column vector of the original point. The resulting column vector gives the coordinates of the transformed point in the transformed space.

In the current section, we are building upon that concept and exploring how to visualize and calculate the transformation of a specific vector using a transformation matrix. Sal uses the notation and concepts already introduced in the previous section to explain this concept.
• In the end, he talks about how you can verify if it ends at the point (10,9) by doing the math. Can I just do that math to find the transformation or no?
• Yes, you can. And it is, generally, easier.
(1 vote)
• When Sal talks about the transformation image, in this case [2/1 2/3] how did he get the image?
(1 vote)
• In the video, the instructor is finding the image of the original vector [3 2] under the linear transformation represented by the transformation matrix:

``[2 1][2 3]``

To find the image, he first applies the transformation matrix to the unit vectors in two dimensions, i.e., the vectors [1 0] and [0 1]. The resulting vectors are the images of the unit vectors under the transformation.

The image of [1 0] is:

``[2 1]``

And the image of [0 1] is:

``[2 3]``

These two vectors form the basis for the transformed space, which is the space that contains all possible images of vectors under this transformation.

To find the image of the original vector [3 2], the instructor expresses it as a linear combination of the transformed unit vectors using the coefficients 3 and 2. That is, he writes:

``[3 2] = 3[2 1] + 2[2 3]``

This equation tells us that the original vector [3 2] can be expressed as a linear combination of the transformed unit vectors.

The image of the original vector is then obtained by adding up the scaled images of the unit vectors, which gives:

``[3 2] (image) = 3[2 1] + 2[2 3] = [10 9]``

So the image of the original vector [3 2] under this transformation is the vector [10 9].
• I don't understand what is meant by the word "image"
(1 vote)
• In linear algebra, the term "image" usually refers to the output of a linear transformation. When we apply a linear transformation to a vector or a set of vectors, we get a new vector or set of vectors as the output, which is called the "image" of the original vector or set of vectors.

In the context of this video, the instructor is using the term "image" to refer to the output obtained by applying the transformation matrix to the unit vectors in two dimensions. These image vectors are then used to find the image of the original vector after the transformation.

So, in short, "image" refers to the output obtained by applying a linear transformation to a vector or a set of vectors.
(1 vote)
• I'm curious about what notation you use when you are doing a transform for matrices
(1 vote)
• When performing a transformation with matrices, I would typically use the standard mathematical notation. The notation for matrix multiplication is the dot product, where the dot represents the multiplication operation. For example, if we have a matrix A and a vector x, we would write the matrix-vector multiplication as Ax.

When we perform a transformation using a matrix, we apply the matrix to the vector that we want to transform. This can be done by multiplying the matrix and the vector together, which gives us a new vector that represents the transformed version of the original vector.

In the video, Sal uses a similar notation for the transformation by writing the image of the original vector as a linear combination of the images of the unit vectors under the transformation matrix. He denotes this with a prime symbol to indicate that it is the transformed vector, and he uses the same dot product notation to compute the transformation.
(1 vote)